Question

An undiscovered planet, many lightyears from Earth, has one moon in a periodic orbit. This moon takes time TT on average to complete one nearly circular revolution around the unnamed planet. If the diameter of the moon's orbit is ????,d, calculate the moon's radial (or centripetal) acceleration ????c.

Answer 1

Answer:

$a_{cp}=\frac{2\pi^2 d}{T^2}$

Explanation:

The equation for centripetal acceleration is

$a_{cp}=\frac{v^2}{r}$

Where v is the tangential velocity and r the radius of the orbit.

To calculate v we notice that the moon travels the circumference of the orbit $C=2\pi r$ in a time T, so we have

$v=\frac{C}{T} =\frac{2\pi r}{T}$

Putting all together:

$a_{cp}=\frac{v^2}{r}=\frac{(\frac{2\pi r}{T})^2}{r}=\frac{4\pi^2 r}{T^2}$

And since the radius r is d/2, where d is the diameter of the moon's orbit, we can write the formula asked:

$a_{cp}=\frac{4\pi^2 r}{T^2}=\frac{4\pi^2 d}{2T^2}=\frac{2\pi^2 d}{T^2}$